MEN OF MATHEMATICS
that; finally every beginner has a great deal of difficulty in
getting noticed here. I have just finished an extensive
treatise on a certain class of transcendental functions [his
masterpiece] to present it to the Institut [Academy of
Sciences], which will be done next Monday* I showed it to
Mr Caucky, but he scarcely deigned to glance at it. And I
dare to say, without bragging, that it is a good piece of
work. I am curious to hear the opinion of the Institut on it.
I shall not fail to share it with you. ...
He then tells what he is doing and continues with a rather
.listurbed forecast of his prospects. 'I regret having set two
years for my travels, a year and a half would have sufficed.' He
lias got all there is to be got out of Continental Europe
und is anxious to be able to devote his time to working up
*7hat he has invented.
So many things remain for me to do, but so long as I am
abroad, all that goes badly enough. If I had my professor-
ship as Mr Kielhau has his! My position is not assured, it
is true, but I am not uneasy about it; if fortune deserts me
in one quarter perhaps she will smile on me in another.
From a letter of earlier date to the astronomer Hansteen we
take two extracts, the first relating to Abel's great project of
re-establishing mathematical analysis as it existed in his day
on a firm foundation, the second showing something of his
human side. (Both are free translations.)
In the higher analysis too few propositions are proved
with conclusive rigour. Everywhere we find the unfortu-
nate procedure of reasoning from the special to the general,
and the miracle is that after such a process it is only seldom
that we find what are called paradoxes. It is indeed exceed-
ingly interesting to seek the reason for this. This reason, in
my opinion, resides in the fact that the functions which
have hitherto occurred in analysis can be expressed for the
most part as powers. ..„ When we proceed by a general
method, it is not too difficult [to avoid pitfalls]; but I have
had to be very circumspect, because propositions without
rigorous proof (i.e. without any proof) have taken root hi
me to such an extent that I constantly run the risk of
using them without further examination. These trifles
will appear in the journal published by Mr Crette.